Method for Generating a Pseudorandom Sequence, and Method for Coding or Decoding a Data Stream

1. A method for coding or decoding a message, the method includes computer instructions implemented in a coding or a decoding module having a processor and a memory, the method comprising: generating a pseudorandom sequence used to code or decode the message using, respectively, the coding module or the decoding module, wherein the pseudorandom sequence used to code the message is the same as the pseudorandom sequence used to decode the coded message, and wherein generating the pseudorandom sequence includes: a) providing a differential equation of initial value x′=f(x,t); b) providing an initial value for the differential equation x 0 =x(t 0 ); c) providing an integration step δ t for the differential equation for time discretization t k =t 0 +k·δ t ,k=1,2,3 . . . ; d) carrying out the numerical integration of the differential equation from the initial value and with the step δ t for obtaining the approximation to solution x k =x(t k ); e) generating a first sequence of values by sampling the values x k numerically representable in floating point in the form of 0.d 0 d 1 d 2 d 3 d 4 . . . d r . . . d w ·10 e , e being the exponent, w the length of the mantissa, d 0 the most representative digit of the mantissa and d r a digit such that it and all the digits to its left of the approximation to solution x k coincide with the exact value of the solution of the differential equation; f) generating the pseudorandom sequence with digits d i . . . d r from a selection of the sequence of values x k wherein i is a predetermined integer value verifying 0<i≦r; g) wherein the pseudorandom sequence is expanded into a sequence with a greater number of elements according to the following steps: pre-establishing a positive integer value DIM; constructing two vectors V 1 and V 2 of dimension DIM of integers from the pseudorandom sequence; constructing an expansion matrix M e of dimension DIM×DIM from the product V 1 ·V 2 T wherein V 2 T is the transposed vector of V 2 ; and generating the expanded sequence by means of concatenating the rows of the matrix M e .

2. The method according to claim 1 , wherein the numerical integration of a system of n differential equations, x′ s =f s (x 1 , x 2 , . . . x n , p 1 , p 2 , . . . p m , t), s=1 . . . n, is carried out in step d), n also being the number of unknowns, and containing m parameters, p j , j=1 . . . m, such that the pseudorandom sequence in step f) is generated from one of the n variables preselected from the system of differential equations.

3. The method according to claim 1 , wherein after step f), each digit d is depicted in binary, with a pre-established word size D1, the concatenation of the digits forming a binary sequence.

4. The method according to claim 1 , wherein after step f), each digit d is made to correspond to a binary expression, the concatenation of the digits forming a binary sequence.

5. The method according to claim 3 , wherein a word size D 2 is pre-established and integer digits are formed from the binary sequence using words of D 2 bits.

6. The method according to claim 1 , wherein: a value K 1 is pre-established; and before generating the expanded sequence each element of the matrix M e is substituted with the value resulting from calculating the K 1 -module thereof.

7. The method according to claim 1 , wherein: in addition to the vectors V 1 and V 2 , a vector V 3 of dimension DIM of integers is constructed from the pseudorandom sequence generated; and on each of the rows of the matrix M e , before generating the expanded sequence by means of concatenating the rows of the matrix M e , rearranging the sequence of the integers in each of the rows of M e by circularly rotating the position of the integers in a pre-established direction, according to the integer value established by the same row of the vector V 3 .

8. The method according to claim 1 , wherein: a value K 2 , preferably the DIM value, is pre-established; and each element of the vector V 3 is substituted with the value resulting from calculating the modulus-K 2 thereof.

9. The method according to claim 1 , wherein for generating the expanded sequence by means of concatenating the rows of the matrix M e only the values of each row are calculated to avoid storing the complete matrix M e .

10. A method for coding a data stream for the transmission of said data by means of a coded stream wherein the coding is the result of comparing the data stream with a second data stream formed by a pseudorandom sequence by means of an exclusive comparison operation (XOR), or a method for decoding a coded data stream wherein the decoding is the result of comparing the coded data stream with a second data stream formed by a pseudorandom sequence by means of an exclusive comparison operation (XOR), characterized in that the generation of the pseudorandom sequence is made by means of a method according to claim 1 .

12. The method according to claim 11 , wherein the exchange channel is encrypted by means of public key.

13. A system for coding messages for transmission on a transmission channel between a first device at a first end of a communication and a second device at a second end of the communication, the system comprising: a message coder at the first device configured to code a message for transmission to, and decoding at, the second device, wherein coding the message using the message coder includes: comparing a pseudorandom coding sequence to the message using an exclusive comparison operation (XOR), wherein the pseudorandom coding sequence is independently generated at the first device and at the second device enabling coding and decoding of the message without transmission of a key; wherein the message coder at the first device is configured to generate the pseudorandom coding sequence by: providing a differential equation of initial value x′=f(x,t); providing an initial value for the differential equation x 0 =x(t 0 ); providing an integration step δ t for the differential equation for time discretization t k =t 0 +k·δ t , k=1,2,3 . . . ; carrying out the numerical integration of the differential equation from the initial value and with the step δ t for obtaining the approximation to solution x k =x(t k ); generating a first sequence of values by sampling the values x k numerically representable in floating point in the form of 0.d 0 d 1 d 2 d 3 d 4 . . . d r . . . d w ·10 e , e being the exponent, w the length of the mantissa, d 0 the most representative digit of the mantissa and d r a digit such that it and all the digits to its left of the approximation to solution x k coincide with the exact value of the solution of the differential equation; generating the pseudorandom coding sequence with digits d i . . . d r from a selection of the sequence of values x k wherein i is a predetermined integer value verifying 0<i≦r, and wherein the pseudorandom coding sequence is expanded into a sequence with a greater number of elements by: pre-establishing a positive integer value DIM; constructing two vectors V 1 and V 2 of dimension DIM of integers from the pseudorandom sequence; constructing an expansion matrix M e of dimension DIM×DIM from the product V 1 ·V 2 T wherein V 2 T is the transposed vector of V 2 ; and generating the expanded sequence by means of concatenating the rows of the matrix M e .

14. A system for decoding coded messages received on a transmission channel established between first device at a first end of a communication and a second device at a second end of the communication, the system comprising: a message decoder at the second device configured to decode a coded message received from the second device, wherein decoding the coded message using the message decoder includes: comparing a pseudorandom coding sequence to the coded message using an exclusive comparison operation (XOR), wherein the pseudorandom coding sequence is independently generated at the first device and at the second device enabling coding and decoding of the message without transmission of a key; wherein the message decoder at the second device is configured to generate the pseudorandom coding sequence by: providing a differential equation of initial value x′=f(x,t); providing an initial value for the differential equation x 0 =x(t 0 ); providing an integration step δ t for the differential equation for time discretization t k =t 0 +k·δ t ·k=1,2,3 . . . ; carrying out the numerical integration of the differential equation from the initial value and with the step δ t for obtaining the approximation to solution x k =x(t k ); generating a first sequence of values by sampling the values x k numerically representable in floating point in the form of 0.d 0 d 1 d 2 d 3 d 4 . . . d r . . . d w ·10 e , e being the exponent, w the length of the mantissa, d 0 the most representative digit of the mantissa and d r a digit such that it and all the digits to its left of the approximation to solution x k coincide with the exact value of the solution of the differential equation; and generating the pseudorandom coding sequence with digits d i . . . d r from a selection of the sequence of values x k wherein i is a predetermined integer value verifying 0<i≦r, and wherein the pseudorandom coding sequence is expanded into a sequence with a greater number of elements by: pre-establishing a positive integer value DIM; constructing two vectors V 1 and V 2 of dimension DIM of integers from the pseudorandom sequence; constructing an expansion matrix M e of dimension DIM×DIM from the product V 1 ·V 2 T wherein V 2 T is the transposed vector of V 2 ; and generating the expanded sequence by means of concatenating the rows of the matrix M e .

15. A communication system for providing secure communication of messages between a first terminal and a second terminal, the system comprising: a message coder at the first terminal configured to generate an encrypted message for transmission to the second terminal by comparing a message with an encryption sequence using an exclusive comparison operation (XOR); a message decoder at the second terminal configured to decrypt the encrypted message received by the first terminal by comparing the encrypted message with a decryption sequence using an exclusive comparison operation (XOR); and wherein the message coder and message decoder are configured to independently generate the encryption and decryption sequences at the first terminal and at the second terminal enabling encrypting and decrypting of the message without transmission of a key, and wherein the encryption and decryption sequences are identical pseudorandom sequences generated at the message coder and at the message decoder by: providing a differential equation of initial value x′=f(x,t); providing an initial value for the differential equation x 0 =x(t 0 ); providing an integration step δ t for the differential equation for time discretization t k =t 0 +k·δ t , k=1,2,3 . . . ; carrying out the numerical integration of the differential equation from the initial value and with the step δ t for obtaining the approximation to solution x k =x(t k ); generating a first sequence of values by sampling the values x k numerically representable in floating point in the form of 0.d 0 d 1 d 2 d 3 d 4 . . . d r . . . d w ·10 e , e being the exponent, w the length of the mantissa, d 0 the most representative digit of the mantissa and d r a digit such that it and all the digits to its left of the approximation to solution x k coincide with the exact value of the solution of the differential equation; and generating the pseudorandom sequence with digits d i . . . d r from a selection of the sequence of values x k wherein i is a predetermined integer value verifying 0<i≦r, and wherein the pseudorandom sequence is expanded into a sequence with a greater number of elements by: pre-establishing a positive integer value DIM; constructing two vectors V 1 and V 2 of dimension DIM of integers from the pseudorandom sequence; constructing an expansion matrix M e of dimension DIM×DIM from the product V 1 ·V 2 T wherein V 2 T is the transposed vector of V 2 ; and generating the expanded sequence by means of concatenating the rows of the matrix M e .

16. The communication system according to claim 15 , wherein the pseudorandom sequences generated at the first terminal and the second terminal are the same.